PROBLEM SOLUTIONS PDF

Preview

Summary

www.soludelibros.blogspot.com Preface This Instructors’ Manual provides solutions to most of the problems in ANTENNAS: FOR ALL APPLICATIONS, THIRD EDITION. All problems are solved for which answers appear in Appendix F of the text, and in addition, solutions are given for a large fraction of the oth...

Description

www.soludelibros.blogspot.com

Preface This Instructors’ Manual provides solutions to most of the problems in ANTENNAS: FOR ALL APPLICATIONS, THIRD EDITION. All problems are solved for which answers appear in Appendix F of the text, and in addition, solutions are given for a large fraction of the other problems. Including multiple parts, there are 600 problems in the text and solutions are presented here for the majority of them. Many of the problem titles are supplemented by key words or phrases alluding to the solution procedure. Answers are indicated. Many tips on solutions are included which can be passed on to students. Although an objective of problem solving is to obtain an answer, we have endeavored to also provide insights as to how many of the problems are related to engineering situations in the real world. The Manual includes an index to assist in finding problems by topic or principle and to facilitate finding closely-related problems. This Manual was prepared with the assistance of Dr. Erich Pacht.

Professor John D. Kraus Dept. of Electrical Engineering Ohio State University 2015 Neil Ave Columbus, Ohio 43210 Dr. Ronald J. Marhefka Senior Research Scientist/Adjunct Professor The Ohio State University Electroscience Laboratory 1320 Kinnear Road Columbus, Ohio 43212

iii

Table of Contents Preface iii Problem Solutions: Chapter 2. Antenna Basics.............................................................................................1 Chapter 3. The Antenna Family...................................................................................17 Chapter 4. Point Sources .............................................................................................19 Chapter 5. Arrays of Point Sources, Part I...................................................................23 Chapter 5. Arrays of Point Sources, Part II .................................................................29 Chapter 6. The Electric Dipole and Thin Linear Antennas .........................................35 Chapter 7. The Loop Antenna .....................................................................................47 Chapter 8. End-Fire Antennas: The Helical Beam Antenna and the Yagi-Uda Array, Part I................................................................................................53 Chapter 8. The Helical Antenna: Axial and Other Modes, Part II..............................55 Chapter 9. Slot, Patch and Horn Antennas ..................................................................57 Chapter 10. Flat Sheet, Corner and Parabolic Reflector Antennas ................................65 Chapter 11. Broadband and Frequency-Independent Antennas.....................................75 Chapter 12. Antenna Temperature, Remote Sensing and Radar Cross Section ............81 Chapter 13. Self and Mutual Impedances ....................................................................103 Chapter 14. The Cylindrical Antenna and the Moment Method (MM).......................105 Chapter 15. The Fourier Transform Relation Between Aperture Distribution and Far-Field Pattern................................................................................107 Chapter 16. Arrays of Dipoles and of Aperture ...........................................................109 Chapter 17. Lens Antennas ..........................................................................................121 Chapter 18. Frequency-Selective Surfaces and Periodic Structures By Ben A. Munk ......................................................................................125 Chapter 19. Practical Design Considerations of Large Aperture Antennas.................127 Chapter 21. Antennas for Special Applications...........................................................135 Chapter 23. Baluns, etc. By Ben A. Munk ..................................................................143 Chapter 24. Antenna Measurements. By Arto Lehto and Pertti Vainikainen ....................................................................................147 Index

153

iv

1 Chapter 2. Antenna Basics 2-7-1. Directivity. Show that the directivity D of an antenna may be written E

,

E

max

,

max

2

r

Z

D 1

,

E

E

4

4

,

2

r d

Z

Solution:

U

D

(

,

)

U

,

U ( , ) m ax

S ( , ) m ax r

2

,

U( , )

U

1 av

2

m

a

x

v

U ( , )d 4

4

S ( , )r

a

,

E

S( , )

,

E

,

Z

Therefore E

,

E

m ax

m ax

r

2

Z

D E

1

area/steradian, so

q.e.d. E

,

4

4

Note that r 2

, ,

2

r d

Z

U

or (watts/steradian) = (watts/meter2)

2

Sr

meter2

2-7-2. Approximate directivities. Calculate the approximate directivity from the half-power beam widths of a unidirectional antenna if the normalized power pattern is given by: (a) P n = cos , (b) P n = cos2 , (c) P n = cos3 , and (d) P n = cosn . In all cases these patterns are unidirectional (+z direction) 90 and P n = 0 for 90 180 . with P n having a value only for zenith angles 0 The patterns are independent of the azimuth angle . Solution: (a) (b)

HP

HP

1

2 cos

(0 .5 )

2

60

o

120

o

,

D

40 , 000 (120 )

2 cos

1

( 0 .5 )

2

45

o

90

o

,

D

2

40 , 000 ( 90 )

2

278

4 . 94

(ans.) (ans.)

2 (c)

HP

2 cos

1

( 3 0 .5 )

2

37.47

o

74 . 93

o

,

40 , 000

D

( 75 )

(ans.)

7 .3

2

2-7-2. continued (d)

HP

2 cos

1

( n 0 .5 )

,

10 , 000

D (cos

1 n

( 0 . 5 ))

2

(ans.)

*2-7-3. Approximate directivities. Calculate the approximate directivities from the half-power beam widths of the three unidirectional antennas having power patterns as follows: P( , ) = P m sin

sin2

P( , ) = P m sin

sin3

P( , ) = P m sin2

sin3

P( , ) has a value only for 0

and 0

and is zero elsewhere.

Solution: To find D using approximate relations, we first must find the half-power beamwidths. HPBW

or

90

HPBW

90

2

2

pattern,

For sin

sin

HPBW

sin 90

1

2 90

HPBW

sin

1

2

For sin2

1

HPBW

,

2

pattern,

sin

2

sin

2

sin

2

90

,

2 1

1 2

HPBW

1

2

sin 90

HPBW

1

2

For sin3

pattern,

sin

,

90

HPBW

,

2

90

o

2

3

sin

3

90

HPBW 2

1 2

,

,

HPBW

120

o

3 HPBW

sin 90

1 3

2

,

HPBW

7 4 .9

o

2

*2-7-3. continued Thus, D

41 , 253 sq. deg. HP

41 , 253

40 , 000

3 . 82

(120 )( 90 )

HP

(ans.)

3 . 70

(120 )( 90 )

41 , 253

4 . 59

(120 )( 74 . 9 )

for P( , ) = sin

40 , 000

4 . 45

(120 )( 74 . 9 )

41 , 253

6 . 12

( 90 )( 74 . 9 )

40 , 000

sin2

(ans.) for P( , ) = sin

(ans.)

5 . 93

for P( , ) = sin2

( 90 )( 74 . 9 )

sin3

sin3

*2-7-4. Directivity and gain. (a) Estimate the directivity of an antenna with this antenna if efficiency k = 0.5.

HP

=2 ,

HP

= 1 , and (b) find the gain of

Solution: (a)

D

40 , 000 HP

(b)

G

kD

HP

40 , 000

2 .0

10

4

or 43.0 dB (ans.)

( 2 )( 1)

0 .5 ( 2 .0

4

10 )

1 .0

10

4

or 40.0 dB (ans.)

2-9-1. Directivity and apertures. Show that the directivity of an antenna may be expressed as

D

E x , y dxdy

4

Ap

E

x , y dxdy

Ap

2

E x, y E

x , y dxdy

Ap

where E(x, y) is the aperture field distribution. Solution: If the field over the aperture is uniform, the directivity is a maximum (= Dm) and the power radiated is P . For an actual aperture distribution, the directivity is D and the power radiated is P. Equating effective powers

4 *

E av E av

Dm P

DP

,

D

P

Dm

4 2

P

Z E x, y E

Ap Ap

Ap x, y

dxdy

Z

2-9-1. continued where

1

E av

therefore

E ( x , y ) dxdy Ap

Ap

E x , y d xd y

4

D

Ap

E

x , y d xd y

Ap

2

E x, y E

q.e.d. x , y d xd y

Ap

E av E av Ap

where

E x, y E

E av E av

x , y d xd y

Ap

1

E av 2

E x, y E

Ae

( E ) av

x , y d xd y

ap

Ap

Ap

2-9-2. Effective aperture and beam area. What is the maximum effective aperture (approximately) for a beam antenna having halfpower widths of 30 and 35 in perpendicular planes intersecting in the beam axis? Minor lobes are small and may be neglected. Solution: 2 A

HP

30

HP

o

o

35 ,

57 . 3

Aem

30

A

o

2

35

2 o

2

3 .1

(ans.)

*2-9-3. Effective aperture and directivity. What is the maximum effective aperture of a microwave antenna with a directivity of 900? Solution:

D

4

Ae m /

2

,

Aem

D 4

2

900

2

71 . 6

2

(ans.)

4

2-11-1. Received power and the Friis formula. What is the maximum power received at a distance of 0.5 km over a free-space 1 GHz circuit consisting of a transmitting antenna with a 25 dB gain and a receiving antenna with a 20 dB gain? The gain is with respect to a lossless isotropic source. The transmitting antenna input is 150 W.

5

Solution: 8

c/ f

3 10 /10

9

0 .3 m ,

2

Dt

Aet

,

Aer

4

2

Dr 4

2-11-1. continued

Pr

Aet Aer

Pt

r

2

Pt

2

2

Dt

2

Dr

2

(4 ) r

2

150

2

316

0 .3

2

100

2

( 4 ) 500

0 . 0108 W

2

10.8 mW

(ans.)

*2-11-2. Spacecraft link over 100 Mm. Two spacecraft are separated by 100 Mm. Each has an antenna with D = 1000 operating at 2.5 GHz. If craft A's receiver requires 20 dB over 1 pW, what transmitter power is required on craft B to achieve this signal level? Solution: 8

c/ f

3 1 0 / 2 .5 1 0

Pr ( req u ired ) Pt

Pr

r

2

Aet

12

100 10

2

2

Pr

2

(4 ) r D

9

10

2

2

Aet

0 .1 2 m , 10

2

4

Aer

4

W 2

Pr

2

D

r (4 ) D

2

2

16

10

2

10

10 (4 ) 10

6

2

0 .1 2

2

10966 W

11 kW

( a n s.)

2-11-3. Spacecraft link over 3 Mm. Two spacecraft are separated by 3 Mm. Each has an antenna with D = 200 operating at 2 GHz. If craft A's receiver requires 20 dB over 1 pW, what transmitter power is required on craft B to achieve this signal level? Solution: 8

c/ f

3 10 / 2 10

Pr

100 10

Pt

Pr

r

2

12

10

2

Aet Aer

10

9

2

(4 ) r D

Aer

2

D 4

W 2

Pr

Aet

0 .1 5 m

2

2

2 2

2

10

10

(4 ) 9 10 4 10

4

12

0 .1 5

2

15 8 W

( a n s.)

6 2-11-4. Mars and Jupiter links. (a) Design a two-way radio link to operate over earth-Mars distances for data and picture transmission with a Mars probe at 2.5 GHz with a 5 MHz bandwidth. A power of 10-19 W Hz-1 is to be delivered to the earth receiver and 10-17 W Hz-1 to the Mars receiver. The Mars antenna must be no larger than 3 m in diameter. Specify effective aperture of Mars and earth antennas and transmitter power (total over entire bandwidth) at each end. Take earth-Mars distance as 6 light-minutes. (b) Repeat (a) for an earth-Jupiter link. Take the earth-Jupiter distance as 40 light-minutes. 2-11-4. continued Solution: (a)

c/ f

Pr (earth ) Pr ( M ars )

8

3 1 0 / 2 .5 1 0

10

19

10

17

5 10

6

5 10

6

Take

Ae ( Mars)

Take

Pt ( Mars)

1 kW

Take

Ae ( earth)

(1/2)

P t ( earth )

P r ( Mars)

P t ( earth )

5

(1/2)

2

1.5

15

11

10

2

9

0 .1 2 m

5 10

13

W

5 10

11

W

3 .5 m

2

350 m

2

2

r

(

ap

0 .5 )

(

ap

0 .5 )

2

Aet ( earth) Aet ( Mars)

( 360

8

3

2

10 ) 0 . 12

3 .5

2

6 . 9 MW

350

To reduce the required earth station power, take the earth station antenna Ae

(1 / 2 )

50

2

3927 m

2

(ans.)

so Pt (e a rth )

Pr ( earth )

Pt ( Mars)

6

6 .9 1 0 (1 5 / 5 0 )

Aet ( Mars) Aer ( earth) r

2

2

2

620 kW

10

( a n s.)

3 .5

3

( 360

3930 8 2

3 10 ) 0 . 12

2

8

10

14

W

which is about 16% of the required 5 x 10 13 W. The required 5 x 10 13 W could be obtained by increasing the Mars transmitter power by a factor of 6.3. Other alternatives

7 would be (1) to reduce the bandwidth (and data rate) reducing the required value of P r or (2) to employ a more sensitive receiver. As discussed in Sec. 12-1, the noise power of a receiving system is a function of its system temperature T and bandwidth B as given by P = kTB, where k = Boltzmann’s constant = 1.38 x 10 23 JK 1. For B = 5 x 106 Hz (as given in this problem) and T = 50 K (an attainable value), P ( noise)

1.38

10

23

50

5

10

6

3 .5

10

15

W

2-11-4. continued The received power (8 x 10 14 W) is about 20 times this noise power, which is probably sufficient for satisfactory communication. Accordingly, with a 50 K receiving system temperature at the earth station, a Mars transmitter power of 1 kW is adequate. (b) The given Jupiter distance is 40/6 = 6.7 times that to Mars, which makes the required transmitter powers 6.72 = 45 times as much or the required receiver powers 1/45 as much. Neither appears feasible. But a practical solution would be to reduce the bandwidth for the Jupiter link by a factor of about 50, making B = (5/50) x 106 = 100 kHz. *2-11-5. Moon link. A radio link from the moon to the earth has a moon-based 5 long right-handed monofilar axial-mode helical antenna (see Eq. (8-3-7)) and a 2 W transmitter operating at 1.5 GHz. What should the polarization state and effective aperture be for the earth-based antenna in order to deliver 10-14 W to the receiver? Take the earth-moon distance as 1.27 light-seconds. Solution: c/ f

8

3 1 0 / 1 .5 1 0

9

0 .2 m ,

From (8-3-7) the directivity of the moon helix is given by D

From Friis formula

12

5

60

and

D Aet ( moon )

4

2

8 Aer

Pr r

2

2

2

Pr ( 4 ) r Pt D

Pt Aet

2

14

10

(3

10

2

2

8